To prove compactness in \(C([0,T];L^p(\mathbb {R}^N))\)  with \(p\in [1,\infty )\) , we employ the well-known Arzelà-Ascoli and Kolmogorov-Riesz compactness theorems (cf. Appendix ). A systematic approach to these theorems are presented in Section . Compared to some previous results in this direction (see e.g. Lemma 2.2 in [1]} and Theorem A.8 in [2]}), we are actually going to deduce equitightness (uniform tail control) for approximate solutions of (REF ). Then we are able to avoid all unnecessary tricks to fulfil the requirements of both Arzelà-Ascoli and Kolmogorov-Riesz, and instead present a minimal and efficient compactness argument. However, we still use the uniform boundedness (\(L^\infty \) -stability) of the solutions to make sure that some of the estimates needed hold. A possible improvement of our results is thus to study (REF ) in a pure \(L^1\) -framework.
